THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.
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This is a reformulation of posting 432: Finite Games and
Incompleteness 4 http://www.cs.nyu.edu/pipermail/fom/2010-June/014847.html
which now supports two separate kinds of chains, which we refer to as
strong chains.
upper shift chains.
PROPOSITION 4.1. In every downward order invariant relation game on
Q^k, the winning set of some (E union {0})^k is weakly upper shift
closed.
PROPOSITION 5.1. In every downward order invariant game on Q^k, the
winning sets of some length r strong chain of finite sets form an
upper shift chain.
Proposition 4.1 with some fixed k, Proposition 5.1 with some fixed k
(r varying), and Proposition 5.1 with some fixed r (k varying), are
provable with large cardinals but not without.
THIS POSTING IS ENTIRELY SELF CONTAINED.
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1. TERMINATING RELATION GAMES AND WINNING SETS.
2. DOWNWARD ORDER INVARIANT GAMES.
3. CHAINS AND THE UPPER SHIFT.
4. INFINITE GAME THEOREM.
5. FINITE GAME THEOREMS.
6. TEMPLATES.
1. TERMINATING RELATION GAMES AND WINNING SETS.
Relation games are (given by) pairs (X,R), where R is contained in X x
X. Alice and Bob alternately play moves, which are elements of X.
Alice starts the game by playing any element of X. Subsequently, each
player must play some y such that x R y, where x is the last move
previously played by their opponent. If a player cannot make a legal
move, then play halts, the player loses, and their opponent wins.
Such a game may never reach adjudication, since a play of the game of
infinite length is generally possible.
A terminating relation game is a relation game such that there are no
plays of the game of infinite length. This is the same as saying that
there is no infinite walk in (X,R).
The winning set for a terminating relation game is the set of all
winning moves.
The notion of winning move in this context is standard from game
theory. Because these games are zero memory, every move is inherently
winning or losing. I.e., the notion of winning/losing move is player
independent.
Let (X,R) be a terminating relation game and A contained in X. The
winning set for A in the game (X,R) is the winning set for the game
(X,R)|A.
The following is well known.
THEOREM 1.1. Let (X,R) be a terminating relation game. We can view
these (X,R) as digraphs with no infinite walk. The winning set for
(X,R) is the unique kernel of (X,R). The winning set for A contained
in X in (X,R) is the unique kernel of the induced subdigraph (X,R)|A.
2. DOWNWARD ORDER INVARIANT GAMES.
We now focus on relation games (Q^k,R). We say that these games are on
Q^k.
We say that (Q^k,R) is downward if and only if x R y implies max(x) >
max(y). Obviously, every downward game on Q^k is terminating.
We say that x,y in Q^k are order equivalent if and only if for all 1
<= i,j <= k, x_i < x_j iff y_i < y_j.
We say that A contained in Q^k is order invariant if and only if for
all order equivalent x,y in Q^k, x in A iff y in A.
An order invariant relation game on Q^k is a relation game (Q^k,R),
where R is an order invariant subset of Q^2k.
3. CHAINS AND THE UPPER SHIFT
A strong chain of subsets of Q^k is a finite or infinite sequence
A_1,...,A_n, where for all 1 <= i < n, every k tuple of consisting of
0 and coordinates of elements of A_i lies in A_i+1.
Let f:Q^k into Q^k. An f chain of subsets of Q^k is a finite or
infinite sequence A_1,...,A_n, where for all 1 <= i < n, A_i union
f[A_i] is contained in A_i+1.
We will focus on the upper shift function, mapping Q^k into Q^k. it is
obtained by adding 1 to all nonnegative coordinates, and keeping all
negative coordinates fixed.
4. INFINITE GAME THEOREM.
PROPOSITION 4.1. In every downward order invariant game on Q^k, the
winning set of some (E union {0})^k is upper shift closed.
THEOREM 4.2. Proposition 4.1 is provable in SRP+ but not from any
consequence of SRP that is consistent with RCA_0. Proposition 4.1
is provably equivalent, over WKL_0, to Con(SRP). These results hold if
we fix k to be any particular sufficiently large integer in
Proposition 4.1.
Here SRP+ = ZFC + "for all k there exists a limit ordinal with the k-
SRP. SRP = ZFC + {there exists a limit ordinal with the k-SRP}_k. The
k-SRP asserts that every partition of the unordered k-tuples from
lambda into two pieces has a homogeneous set that is a stationary
subset of lambda.
EFA is exponential function arithmetic.
5. FINITE GAME THEOREMS.
PROPOSITION 5.1. In every downward order invariant game on Q^k, the
winning sets of some length r strong chain of finite sets form an
upper shift chain.
Note that Proposition 5.1 is explicitly Pi02.
The norm of a finite sequence of finite subsets of Q^k, is the maximum
of the magnitudes of the denominators and numerators in the reduced
form of the coordinates of the elements of its terms.
PROPOSITION 5.2. In every downward order invariant game on Q^k, the
winning sets of some length r strong chain of norm at most (8kr)!!
form an upper shift chain.
Note that Proposition 5.2 is explicitly Pi01.
Here (8kr)!! is a safe and convenient upper bounds, which
can be sharply reduced.
THEOREM 5.3. Propositions 5.1, 5.2 are provable in SRP+, but not from
any consequence of SRP that is consistent with EFA. Propositions 5.1,
5.2 are provably equivalent, over EFA,
to Con(SRP). These results hold if we fix k to be any particular
sufficiently large integer in these Propositions. These results also
hold if we alternatively fix r to be any particular sufficiently large
integer in these Propositions.
6. TEMPLATES.
Let PL(Q^k) be the class of all piecewise linear functions from Q^k
into Q^k.
TEMPLATE. Let f be in PL(Q^k). In every downward order invariant game
on Q^k, the winning set of some (E union {0})^k is f closed.
TEMPLATE. Let f be in PL(Q^k). In every downward order invariant game
on Q^k, the winning sets of some length r strong chain of finite sets
form an f chain.
The plan is to determine the status of all instances of these Templates.
It should be easier to do this for the class DPL(Q^k) of diagonal
piecewise linear functions from Q^k into Q^k. These are given by the
action of a single piecewise linear function from Q into Q. Note that
our upper shift map from Q^k into Q^k lies in DPL(Q^k).
We can allow partial piecewise linear functions, or even finite lists
of (partial) piecewise linear functions such functions, for more
powerful Templates.
**********************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 433rd in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-349 can be found athttp://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.
350: one dimensional set series 7/23/09 12:11AM
351: Mapping Theorems/Mahlo/Subtle 8/6/09 10:59PM
352: Mapping Theorems/simpler 8/7/09 10:06PM
353: Function Generation 1 8/9/09 12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1 8/9/09 6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2 8/10/09 6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem 8/14/09 9:31AM
357: HIGH SCHOOL Games/Update 8/20/09 10:42AM
358: clearer statements of HIGH SCHOOL Games 8/23/09 2:42AM
359: finite two person HIGH SCHOOL games 8/24/09 1:28PM
360: Finite Linear/Limited Memory Games 8/31/09 5:43PM
361: Finite Promise Games 9/2/09 7:04AM
362: Simplest Order Invariant Game 9/7/09 11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1 9/24/09 1:06PM
366: Free Reductions and Large Cardinals/polished 9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals 10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement 10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09 12:14PM
371: Vector Reduction and Large Cardinals 11/21/09 1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09 5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals 12/7/09 9:17AM
374: Upper Shift Greedy Chain Games 12/12/09 5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09 2:23PM
377: The Polynomial Shift Theorem 12/25/09 2:39PM
378: Upper Shift Clique Sequences and Large Cardinals 12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems 12/28/09 7:06AM
381: Trigonometric Shift Theorem 12/29/09 11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals 12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09 3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences 1/1/10 7:35PM
386: Terrifically and Extremely Long Finite Sequences 1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos 1/6/10 10:41PM
388: Goedel's Second Again/definitive? 1/7/10 11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1 3/2/10 7:30PM
395: Free Reduction Theory 2 3/7/10 5:41PM
396: Free Reduction Theory 3 3/7/10 11:30PM
397: Free Reduction Theory 4 3/8/10 9:05AM
398: New Free Reduction Theory 1 3/10/10 5:26AM
399: New Free Reduction Theory 2 3/12/10 9:36AM
400: New Free Reduction Theory 3 3/14/10 11:55AM
401: New Free Reduction Theory 4 3/15/10 4:12PM
402: New Free Reduction Theory 5 3/19/10 12:59PM
403: Set Equation Tower Theory 1 3/22/10 2:45PM
404: Set Equation Tower Theory 2 3/24/10 11:18PM
405: Some Countable Model Theory 1 3/24/10 11:20PM
406: Set Equation Tower Theory 3 3/25/10 6:24PM
407: Kernel Tower Theory 1 3/31/10 12:02PM
408: Kernel tower Theory 2 4/1/10 6:46PM
409: Kernel Tower Theory 3 4/5/10 4:04PM
410: Kernel Function Theory 1 4/8/10 7:39PM
411: Free Generation Theory 1 4/13/10 2:55PM
412: Local Basis Construction Theory 1 4/17/10 11:23PM
413: Local Basis Construction Theory 2 4/20/10 1:51PM
414: Integer Decomposition Theory 4/23/10 12:45PM
415: Integer Decomposition Theory 2 4/24/10 3:49PM
416: Integer Decomposition Theory 3 4/26/10 7:04PM
417: Integer Decomposition Theory 4 4/28/10 6:25PM
418: Integer Decomposition Theory 5 4/29/10 4:08PM
419: Integer Decomposition Theory 6 5/4/10 10:39PM
420: Reduction Function Theory 1 5/17/10 2:53AM
421: Reduction Function Theory 2 5/19/10 12:00PM
422: Well Behaved Reduction Functions 1 5/23/10 4:12PM
423: Well Behaved Reduction Functions 2 5/27/10 3:01PM
424: Well Behaved Reduction Functions 3 5/29/10 8:06PM
425: Well Behaved Reduction Functions 4 5/31/10 5:05PM
426: Well Behaved Reduction Functions 5 6/2/10 12:43PM
427: Finite Games and Incompleteness 1 6/10/10 4:08PM
428: Typo Correction in #427 6/11/10 12:11AM
429: Finite Games and Incompleteness 2 6/16/10 7:26PM
430: Finite Games and Incompleteness 3 6/18/10 6:14PM
431: Finite Incompleteness/Combinatorially Simplest 6/20/10 11:22PM
432: Finite Games and Incompleteness 4 6/26/10 8:39PM
Harvey Friedman